BANDAR SERI BEGAWAN, Brunei—President Barack Obama's absence from summits in Southeast Asia this week was expected to open up the floor for an increasingly assertive China, but the spotlight turned more on another leader eager to bolster his relations with Southeast Asia and extend his country's reach: Japanese Prime Minister Shinzo Abe.

Analysts said that without Mr. Obama, there were fewer heads of state with the standing to raise important security issues. Secretary of State John Kerry filled in at the Asia-Pacific...